The importance of "relying on God" is a common topic in Christian parlance, easy to dismiss as cliche, difficult to understand, more difficult to practice. What does it mean to rely on God and not on one's self.
I see two broad components to the concept. The first is spiritual, the second practical. Christianity teaches that all mankind is sinful, living in a state of disobedience toward God, and as a result, separation from him. It is God's desire to resolve this separation, and he must eradicate mankind's sin in order to do so. To rely on God is to accept that it is God and not man who atones for and eradicates sin through the death and resurrection of Christ his Son. None of man's good works, earnest remorse or cleansing rituals is sufficient to purge man of the practical habits or spiritual desolation of sin. We learn to rely on God first for salvation.
This reliance should manifest itself in a new way of practical living. To me, this sort of reliance is a much more difficult concept. Whatever my spiritual state, aren't my legs carrying me, my hands cleaning, my work providing the money for my needs? I see three major practices by which Christians rely on God. The first is prayer for all of one's needs. Jesus' model prayer in Matthew 6 includes the request "Give us this day our daily bread," confirming both that it is proper to pray for one's basic needs and that one must do so regularly. It recognizes that even if our circumstances allow us to provide for ourselves, God is the author of those circumstances. The second practice is to avoid worry, as Jesus instructs shortly thereafter. This is a difficult practice, seemingly requiring an impossible mastery of one's thoughts. However, one can certainly recognize oneself worrying, and in those circumstances, focus one's attention elsewhere. Finally, one must not use dishonest means to overcome difficult circumstances.
As individuals describe it, "relying on God" often sounds like a mystical experience, the result of a mysterious sensation of assurance of God's provision. I do not want to deny the possibility of such experiences. But more importantly, I want to emphasize the practical aspects of this idea.